Let's continue our discussion of the subsystems in the impedance meter that we've been discussing in the previous posts (see "Related Posts" at the end of page 3). We now move on to the next to last major functional block: the phase comparator or detector.
This block also offers a variety of implementation choices, and all of them can be integrated monolithically. A simple reactance-measuring meter would not need a phase detector, and the bridge equations presented in the first article (Z Meter on a Chip? Impedance Meter Bridge Circuits), which are based on a simple reactance model of the unknown to be measured, Zx, would suffice.
While reactance measurement (or measurement of L, C, and R) is the main objective, Z meters are expected to also acquire values of circuit elements of more complicated models for Zx based on the decomposition of impedance into its real (or resistive) and imaginary (reactive) components, as shown in the impedance diagram below.
ZL is the impedance of an inductor with parasitic series resistance, RL. ZC is the impedance of a capacitor with series resistance RC. Both of these parasitic values are of interest and can be acquired by decomposing Zx into these components.
The magnitude of Zx is derived directly from the bridge variables by Ohmís Law. The phase angle can be measured in the time domain by measuring the time intervals between zero-crossings of the two sine waveforms. This is an attractive method when a μC with a high-resolution counting capability is involved.
Otherwise, the phase is determined in the frequency domain using a phase detector: a phase comparator followed by a low-pass filter. The comparator multiplies vvx for L or vix for C by the generator sine-waves that are 0° (I, or in-phase) and 90° (Q, or quadrature) relative to the vg- stabilized quantity (vix for L, vvx for C). The conceptually simplest form of detector multiplies sine-wave by sine-wave and then integrates the result over the cycle. The integration or low-pass filtering can be implemented multiple ways. The simplest is a passive RC integrator. A more elegant choice is a synchronous integrator.
The multiplier as a phase comparator is expressed in its equations. Let the bridge quantity to be compared be vx and the reference sine-wave derived from the generator waveform(s) be vg. Then the bridge current through measured impedance Zx has an amplitude of Ig and
θx is the phase of vx relative to vg. The cosine with a difference of angles is expanded so that
The generator sine-wave to be multiplied by vx is scaled for a desired output current, Io, to be multiplied by the impedance component being measured. The generator input amplitude to the multiplier is removed by dividing it out using the factor, Ig/2. Then what is multiplied by vx is a unitless quantity,
and results in
Following the multiplier is an integrator that synchronously integrates over the cycle. The integrals are cycle averages;
What these equations tell us is that in-phase waveforms (sin and sin, or cos and cos) result in a cycle-averaged output of ½ while sin and cos, having a 90° phase difference (in quadrature), average to zero.